On Khovanov local equivalence groups

Abstract

Inspired by work in Seiberg--Witten and Heegaard Floer homology, Dunfield, Schütz, and the author constructed an abelian "local equivalence group" built from pairs of chain complexes, and a homomorphism from the smooth concordance group to this group that uses the odd Khovanov complex and the Bar-Natan deformation of the even Khovanov complex. Lewark independently gave a related local equivalence group and concordance invariant, using only the Bar-Natan deformation. In this paper, we construct (ℤ/2ℤ)^∞ subgroups of Lewark's local equivalence group and the kernel of the map from our local equivalence group to Lewark's, as well as ℤ^∞ direct summands of Lewark's group and this kernel, invisible to the s-invariants. The construction gives at least six ℤ-valued concordance homomorphisms linearly independent from s, τ, and Dai--Hom--Stoffregen--Truong's invariants.

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BibTeX RIS

BibTeX

@misc{lipshitz2026khovanov,
  title = {{On Khovanov local equivalence groups}},
  author = {Robert Lipshitz},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37595},
  eprint = {2610.37595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37595},
}

RIS

TY  - UNPB
TI  - On Khovanov local equivalence groups
AU  - Robert Lipshitz
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37595
UR  - https://arxiv.org/abs/2610.37595
AB  - Inspired by work in Seiberg--Witten and Heegaard Floer homology, Dunfield, Schütz, and the author constructed an abelian "local equivalence group" built from pairs of chain complexes, and a homomorphism from the smooth concordance group to this group that uses the odd Khovanov complex and the Bar-Natan deformation of the even Khovanov complex. Lewark independently gave a related local equivalence group and concordance invariant, using only the Bar-Natan deformation. In this paper, we construct (ℤ/2ℤ)^∞ subgroups of Lewark's local equivalence group and the kernel of the map from our local equivalence group to Lewark's, as well as ℤ^∞ direct summands of Lewark's group and this kernel, invisible to the s-invariants. The construction gives at least six ℤ-valued concordance homomorphisms linearly independent from s, τ, and Dai--Hom--Stoffregen--Truong's invariants.
ER  -

Abstract

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